Let $X$ be a locally compact Hausdorff space and $X_\infty = X \cup \{ \infty \}$ its one-point compactification. We identify $C(X_\infty) = C_0(X) \oplus \mathbb{R}$ by mapping $g \in C(X_\infty)$ to $(g|_X - g(\infty), g(\infty))$. By duality we can then identify $(C_0(X) \oplus \mathbb{R})' = C(X_\infty)'$ isometrically isomorphic as Banach spaces. Since the left-hand side is a finite direct sum of Banach spaces we get the identifications $C_0(X)' \oplus \mathbb{R} = C_0(X)' \oplus \mathbb{R}' = (C_0(X) \oplus \mathbb{R})' = C(X_\infty)'$. From the Riesz representation theorem, we then get
$$M(X) \oplus \mathbb{R} = M(X_\infty)$$
where $M(Y)$ is the space of signed Radon measures on a locally compact space $Y$.
On the other hand, we can also look at the Lebesgue decomposition for measures $\nu \in M(X_\infty)$ w.r.t. $\delta_\infty$ which gives $\nu = \nu_s + \nu_{ac} = \nu_s + c \delta_\infty$ for some $c \in \mathbb{R}$, where $\nu_s \in Rca(X_\infty)$ is the singular part $\nu_s \perp \delta_\infty$ and $\nu_{ac} = c \delta_\infty$ the absolutely continuous part. Since $\lVert \nu \rVert = \lVert \nu_s \rVert + \lVert \nu_{ac} \rVert$ we get a (topological) direct sum decomposition
$$M(X_\infty) = M(X_\infty)_s \oplus \mathbb{R}.$$
I would like to know whether these two identifications of $M(X_\infty)$ are in fact identical, i.e. that these two identifications of $M(X)$ imply an identification $M(X_\infty)_s = M(X)$ (and that of the two one-dimensional subspaces $\mathbb{R}$).